I want to prove that a connected graph with m edges and n vertices must have at least one vertex of odd degree. In particular, I want to prove this for a graph of 53 edges and 11 vertices; but also in general, is there a way to prove this for a more general case, given a set of parameters for m and n?
As already shown, it's not true in general (any simple cycle has every node of even degree).
The case with 53 edges and 11 vertices is particular, since 11*10/2=55, so it's a complete graph without 2 edges, and it's easy to prove that there exists a node with odd degree, since there are basically only 2 cases.
But this reasoning fails very often. For example there exists a graph with 11 vertices and 52 edges that has no nodes of odd degree (try to prove it by yourself ;) )